Triaxial leak criterion with quadratic dependence on effective pressure for optimizing threaded connections in well tubulars

ABSTRACT

Example systems, devices, and methods are described for evaluating the relative leak safety of threaded connections in well tubulars. The method includes evaluating leak risk using a quadratic leak criterion, which is a function of the effective pressure (Pe) and includes three constants: a first constant called the leak path factor (δ), a thread modulus (α), and a makeup leak resistance (β). The method in some implementations includes identifying at least three test cases with different Δp expected to leak under certain conditions. The method includes fitting a quadratic expression to a subset of values associated with the leaking test cases, in which the resulting quadratic expression includes values for the three constants. The quadratic leak criterion can be expressed as a leak safety factor for evaluation of particular threaded connections, load cases, and conditions, thereby optimizing the design and selection of threaded connections and also optimizing connection performance during drilling and production operations.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit of and priority to U.S.Provisional Application 63/263,032 filed Oct. 26, 2021, entitled,“Triaxial Leak Criterion for Connections Based on Leak Limit withHydrostatic Dependence.” The present application is acontinuation-in-part of currently pending U.S. application Ser. No.17/452,064 filed Oct. 23, 2021, which is a continuation-in-part of U.S.application Ser. No. 16/413,331 filed May 15, 2019 (now U.S. Pat. No.11,156,256), which claims the benefit of and priority to U.S.Provisional Application 62,671,771 filed May 15, 2018. The entirety ofeach identified application is incorporated herein by reference.

TECHNICAL FIELD

Examples set forth in the present disclosure relate to the design ofwell tubulars and the optimization of threaded connections in welltubulars. More particularly, but not by way of limitation, the presentdisclosure describes using a quadratic leak criterion with threeconstants and an associated leak safety factor to evaluate the relativesafety of threaded connections.

BACKGROUND

During the drilling process, a well hole segment is typically supportedby a tubular pipe called a casing. The next casing inside the previouscasing supports the next hole segment, and so on, until the casingreaches the total depth of the well. The innermost pipe is theproduction string called tubing. The term ‘tubular’ will be used herefor all wellbore pipes, including tubing, casing, liners, and tiebacks,as well as drill pipe, bottom hole assemblies, and work strings. In moststrings, all the joints and connections are the same; namely, the samepipe diameters (inner and outer), the same connection type and size, andthe same yield strength. Tapered strings have two or more sections inwhich all the joints and connections are the same.

Most well tubulars are fabricated in cylindrical sections (typically,thirty to forty feet in length), oriented in a column, and joinedtogether with threaded connections. The size, wall thickness, and yieldstrength of the tubular and its threaded connections are structurallydesigned to reduce the risk of failure, including pipe-body failure(burst, collapse, or axial parting) and threaded connection failure(leak, pull-out, or thread fracture).

For pipe body design, the yield strength (Y) of the pipe material(steel, typically) can be measured by placing a rod-shaped specimen ofthe material under axial tension until it yields. To evaluate pipeintegrity, the yield point of the pipe body is typically compared to thevon Mises stress (σ_(VM)), which is a scalar quantity determined fromthe stresses acting on the pipe body. In general, if the von Misesstress (σ_(VM)) is less than the yield strength (Y), the pipe body issafe.

For connection design, connection failure due to leak is an importantpart of current tubular design in the petroleum industry. The threadedconnections currently available include standard connections (sometimescalled API connections, referring to the standards of the AmericanPetroleum Institute) and proprietary designs sometimes called premiumconnections. The mechanical behavior of a threaded connection can beestimated by using finite element analysis (FEA) or by conductingextensive testing. Current qualification testing procedures for leakassessment of a threaded connection involve numerous laboratory tests,conducted using specified axial forces (tension or compression) andspecified internal and external pressures. The test procedure isrepeated for a variety of different conditions, using multiple specimensof the connection. Each specimen is tested until it leaks, or until thepipe (or the connection) fails. Internal leak tests are performed withno external backup pressure. External leak tests are performed with nointernal backup pressure. The total cost of such testing can exceed onemillion dollars—and the results apply only to the specific threadedconnection tested, and only to the precise size tested. Additional FEAor laboratory testing must be conducted for other sizes of the sameconnection design. The test results are typically plotted on a graph todisplay a connection usage envelope; analogous to the von Mises ellipse,with pressure on the ordinate (internal pressure positive) and axialforce on the abscissa (tension positive). If the estimated loads on athreaded connection fall within the leak envelope, the connection issafe.

For tubular design, including the design for threaded connections, thecurrently used design criteria do not include a consideration ofhydrostatic pressure. The current failure theories and equations forboth the pipe body and for connections are based on shear and ignorehydrostatic pressure.

BRIEF DESCRIPTION OF THE DRAWINGS

Features of the various implementations disclosed will be readilyunderstood from the following detailed description, in which referenceis made to the appending drawing figures. A reference numeral is usedwith each element in the description and throughout the several views ofthe drawing. When a plurality of similar elements is present, a singlereference numeral may be assigned to like elements, with an addedlower-case letter referring to a specific element.

The various elements shown in the figures are not drawn to-scale unlessotherwise indicated. The dimensions of the various elements may beenlarged or reduced in the interest of clarity. The several figuresdepict one or more implementations and are presented by way of exampleonly and should not be construed as limiting. Included in the drawingsare the following figures:

FIG. 1 is a graphical representation of a triaxial connection leakcriterion, plotted as a leak line;

FIG. 2 is a graphical representation of a triaxial connection leakcriterion, plotted as a leak circle;

FIG. 3 is a graphical representation of a leak safety factor for aparticular string, including a load case line for a number of loadcases;

FIG. 4 is a cross-sectional view of an example tubing or casing coupledconnection having a radial interference seal;

FIG. 5 is a cross-sectional view of an example drill string connectionhaving an axial interference shoulder seal;

FIG. 6 is a graphical representation of the predicted leak conditionsfor an example API LTC connection;

FIG. 7 is a graphical representation of the predicted failure conditionsfor a pipe-body alone and for the example API LTC connection;

FIG. 8 is a partial cross-sectional view of an example generic premiumconnection;

FIG. 9 is a table of values associated with a plurality of exampleleaking test cases;

FIG. 10 is a graph of the leaking test cases from FIG. 9 and a quadraticexpression fitted to the values; and

FIG. 11 is a graphical representation of a quadratic leak safety factorfor an example string, including example load case lines for a number ofexample load cases.

DETAILED DESCRIPTION

Various implementations and details are described with reference to anexample method of calculating a leak safety factor associated with athreaded connection as a function of effective pressure (Pe=P−σn) andvon Mises stress (σ_(VM)). The leak safety factor includes a quadraticexpression with three constants: a leak path factor (δ), a threadmodulus (α), and a makeup leak resistance (β). The method includesapproving the threaded connection if the leak safety factor is greaterthan a threshold value. In some implementations, the method includesgenerating at least three leaking test cases with different Δp=Pi−Po,each characterized by a combination of loads expected to result in leakat the threaded connection, to determine the three leak constants. Foreach leaking test case, the method includes calculating a subset ofvalues including an effective pressure value (Pe) and a von Mises stressvalue (σ_(VM)). The method includes fitting a quadratic expression tothe subset of values. The resulting quadratic expression includes valuesfor the three constants: the leak path factor (δ), the thread modulus(α), and the makeup leak resistance (β).

The following detailed description includes numerous details andexamples that are intended to provide a thorough understanding of thesubject matter and its relevant teachings. Those skilled in the relevantart may understand how to apply the relevant teachings without suchdetails. This disclosure is not limited to the specific devices,systems, and methods described because the relevant teachings can beapplied or practiced in a variety of ways. The terminology andnomenclature used herein is for the purpose of describing particularaspects only and is not intended to be limiting.

Those skilled in the relevant art will recognize and appreciate thatmany changes can be made to the various aspects of the implementationsdescribed herein, while still obtaining the beneficial results. It willalso be apparent that some of the desired benefits can be obtained byselecting some of the features, but not others. Accordingly, those whowork in the art will recognize that many modifications and adaptationsare possible and may even be desirable in certain applications, and thatthese are part of the disclosure.

The terms “comprising” and “including,” and any forms thereof, areintended to indicate a non-exclusive inclusion; that is, to encompass alist that includes the items listed and may include others not expresslylisted. As used herein, the singular forms “a,” “an” and “the” includeplural referents unless the context clearly dictates otherwise. Thus,for example, reference to a component can include two or more suchcomponents unless the context indicates otherwise. Relational terms suchas “first” and “second” and the like may be used solely to distinguishone element or action from another, without implying any particularorder between such elements or actions.

The terms “optional” or “optionally” mean that the subsequentlydescribed element or action may or may not occur. In other words, such adescription includes instances where the element or action occurs andinstances where it does not.

Then term “facilitate” means to aid, assist, enable, improve, or makeeasier. The term “inhibit” means to hinder, restrain, impede, restrain,thwart, oppose, or obstruct.

The words “proximal” and “distal” are used to describe items or portionsof items that are situated closer to and away from, respectively, a useror a viewer. Thus, for example, the near end or other portion of an itemmay be referred to as the proximal end, whereas the generally opposingportion or far end may be referred to as the distal end.

As used herein, the term “constant” refers to a quantity or value thatrepresents an inherent property of an object or thing. As opposed to avariable, a constant associated with an object does not change inresponse to extrinsic states or conditions. For example, the yieldstrength (Y) of a particular material is a constant.

Currently, the role of hydrostatic pressure in the design of welltubulars is not fully appreciated in the current failure theories foryield, buckling, connection leak, pull-out, or thread fracture.Hydrostatic pressure refers to the pressure exerted by a fluid orfluid-like substance on a body immersed in the substance. The currentlyused design criteria, failure theories, and equations for tubular designdo not depend on hydrostatic pressure.

The total stress at each point in a tubular can be divided into a shearpart and a hydrostatic part. The von Mises stress, defined as a scalarmeasure of the shear part, is used to assess the yield of the pipematerial or, more specifically, to assess burst, collapse, tensilefailure, and compressive failure of the pipe-body. The Lubinskifictitious force, often called the effective force, is used to evaluatebuckling. Both yield and buckling are independent of the hydrostaticpart of the stress; i.e., shear behavior alone governs these designcriteria.

As part of tubular design, threaded connections, especially proprietaryconnections, are often evaluated and qualified for strength and sealingwith design plots, called connection usage or service envelopes. Theseenvelopes are used similarly to the von Mises ellipse and have the sameaxes, namely axial force and differential pressure. Laboratory testsand/or FEA (finite element analysis) are performed to develop theenvelopes. The leak tests are typically performed with no backuppressure and with loads up to some percent of von Mises yield of thepipe body, so the effect of hydrostatic pressure is not even considered,implying that connection leak depends only on the shear part of thestress, like the pipe body.

The leak envelope for a threaded connection based on laboratory testresults, if available, is used for tubular design by assuming that theinternal or external pressure can be replaced with the differentialpressure. This assumption is not supported by experimental evidence,creates inaccuracy, and eliminates the effect of hydrostatic pressure.Because the hydrostatic effect is disregarded, current connection designis biaxial (at best).

NEW APPROACH: Contrary to current practice, and as demonstrated herein,the leak resistance of a threaded connection is directly dependent onhydrostatic pressure. Hence, leak resistance is a function of thethreaded connection's location (depth) in the well string. Tubularanalysis using the von Mises equations alone (which assume shear only)does not adequately characterize the risk of leak at a threadedconnection.

A new triaxial connection leak criterion is described herein, whichincludes the effect of hydrostatic pressure in a linear equation. Basedon the new leak criterion, a new leak connection safety factor isdescribed for evaluating connections and designing tubulars. A leak lineand a leak circle are also described, which provide a graphicalrepresentation to quickly identify whether the expected loads arecritical and will cause leak.

The tubular stresses can be calculated at every depth (z) based on theinternal pressure (Pi), external pressure (Po), and axial force (Fz).(For clarity in small print, these and other denoted variables mayappear herein without subscript letters.) This set ofconditions—internal pressure (Pi), external pressure (Po), and axialforce (Fz)—is known or can be measured or determined, in most cases,including any effect from temperature. As used herein, the depth (z)represents the axial or longitudinal direction along a tubular string,whether the well is solely vertical or not. For a vertical well, themeasured depth equals the true vertical depth. For a non-vertical well,the measured depth is greater than the true vertical depth.

The complete description of the stress acting at a material point on abody is a nine-component tensor, which can be expressed as athree-by-three matrix (often depicted as a cube, with one normal stressand two shear stresses on each of the six faces; the stresses onopposing faces are equal). For a well tubular, in cylindricalcoordinates, the normal stresses include the radial stress (σr), thehoop stress (σθ), and the axial stress (σz). The radial stress and hoopstress depend only on the internal pressure (Pi), external pressure(Po), and the inner and outer diameter of the pipe, and can becalculated using the Lame equations. The axial stress can be calculatedfrom the axial force (Fz), which includes pipe weight, and thecross-sectional area of the pipe. Most tubular design does not considershear stresses, such as the stress generated by friction between theouter pipe surface and the borehole wall.

HYDROSTATIC PRESSURE: The leak resistance of a threaded connection isdirectly dependent on hydrostatic pressure. The mean normal stress, P,which is defined as the scalar measure of the hydrostatic part of thestress, is equal to the average of the three normal stresses, asfollows, in cylindrical coordinates for a tubular:

$P = {\frac{\sigma_{r} + \sigma_{\theta} + \sigma_{z}}{3}.}$

In fact, by definition, the sum of the three normal stresses is simplythe first stress invariant, which itself is a scalar and independent ofthe coordinate system. This coordinate independence (just as with thevon Mises stress, which is the second invariant of the shear part of thestress) is an important feature of the new triaxial leak criterion. Insolid mechanics and as used herein, tensile stress is consideredpositive and compressive stress is negative. The term ‘hydrostaticpressure’ as used herein is equivalent to negative of the mean normalstress, P, which reduces to the conventional meaning of hydrostaticpressure in certain circumstances; e.g., at the bottom of an open-endtubular in a drilled hole filled with static fluid, all three normalstresses in the tubular at its base are compressive and equal to thenegative of the fluid pressure at that depth. In most cases, the threenormal stresses at any point along the string and within the tubularwall are not equal. Hence, the mean normal stress, P, is triaxial andgeneralizes the hydrostatic concept to three dimensions.

TRIAXIAL CONNECTION LEAK CRITERION: The leak behavior of a threadedconnection under triaxial stress is fully described by the newconnection leak criterion 25:

σ_(VM) =−αP+β  “Triaxial Leak Criterion 25”

where P is the mean normal stress, and where alpha (α) and beta (β) arenew connection constants, as described and defined herein. The meannormal stress, P, in the Triaxial Leak Criterion 25 introduces thedependence of leak on hydrostatic pressure—which is fundamental anddifferent from all prior leak theories.

The new Triaxial Leak Criterion 25 is also the basis for a new triaxialleak connection safety factor:

${SF_{Leak}} = \frac{{{- \alpha}P} + \beta}{\sigma_{VM}}$

in which the numerator represents the working limit and the denominatorrepresents the working stress. In practice, as described herein, adesigner can approve a threaded connection if the leak safety factor isgreater than a threshold value, which is typically a number equal to orgreater than one. An additional ‘design factor’ may be selected by thedesigner (or imposed by an operator) as an additional margin of safety,in order to account for manufacturing tolerances and the like. Forexample, a designer or operator may specify 1.25 for the threshold value(which is the number typically used for the pipe body).

The connection constants, alpha (α) and beta (β), represent inherentproperties of a particular threaded connection, independent of extrinsicstates or conditions. The connection constants for a particularconnection are analogous to the material constant for a particularmaterial; for example, the yield strength constant (Y) for a pipe-bodymaterial (steel, typically). In fact; if alpha (α) equals zero, thehydrostatic dependence is eliminated and the Triaxial Leak Criterionreduces identically to the von Mises criterion (similar to that for thepipe body), and the constant beta (β) is analogous to the yield strength(Y).

Using the known equations (without shear) for von Mises stress (σ_(VM))and for mean normal stress (P), the Triaxial Leak Criterion becomes:

$\sigma_{VM} = {\sqrt{\frac{1}{2}\left\lbrack {\left( {\sigma_{r} - \sigma_{\theta}} \right)^{2} + \left( {\sigma_{\theta} - \sigma_{z}} \right)^{2} + \left( {\sigma_{z} - \sigma_{r}} \right)^{2}} \right\rbrack} = {{{- \alpha}\frac{\left( {\sigma_{r} + \sigma_{\theta} + \sigma_{z}} \right)}{3}} + {\beta.}}}$

Using the expressions for the axes of the von Mises circle and theLubinski formula for neutral stress (σ_(n)), the Triaxial Leak Criterionbecomes:

$\sqrt{(\Delta)^{2} + \left( {\Delta\sigma}_{z} \right)^{2}} = {{{- \alpha}\frac{{2\sigma_{n}} + \sigma_{z}}{3}} + \beta}$

where ΔP is the y-axis and Δσ_(z) is the x-axis of the von Mises circle,and where the mean normal stress (P) is expressed as a function of theLubinski neutral stress (σ_(n)) and the axial stress (σz).

Adding and subtracting the Lubinski neutral stress (σ_(n)) in the firstterm on the right, defining Δσ_(z) as Lubinski's “excess axial stressabove its neutral value (σ_(n)),” and re-arranging terms, the TriaxialLeak Criterion becomes:

${\sqrt{(\Delta)^{2} + \left( {\Delta\sigma_{z}} \right)^{2}} + {\frac{\alpha}{3}\left( {\Delta\sigma_{z}} \right)}} = {{{- \alpha}\sigma_{n}} + {\beta.}}$

Using the equation for the Lubinski neutral stress (σ_(n)), which is afunction of the internal radius (a), external radius (b), internalpressure (Pi) and external pressure (Po), the Triaxial Leak Criterionbecomes:

${\sqrt{(\Delta)^{2} + \left( {\Delta\sigma_{z}} \right)^{2}} + {\frac{\alpha}{3}\left( {\Delta\sigma_{z}} \right)}} = {{\alpha\frac{{b^{2}p_{o}} - {a^{2}p_{i}}}{b^{2} - a^{2}}} + {\beta{``{{Leak}{Load}{Equation}}"}}}$

where the internal radius (a) and external radius (b) refer to thedimensions of the threaded connection, not the pipe body. For alphaequals zero, the Leak Load Equation reduces to the equation for the vonMises circle. If the internal pressure (Pi) is equal to the externalpressure (Po) and is equal to the pressure exerted by the drilling mudat a depth (mud density (ρ) times depth (z)), then the right side of theequation above becomes (α) (ρ) (z)+β. This demonstrates that leakresistance is dependent on depth (z). Finally, the Leak Load Equationalso provides the basis for a graphical representation for leakassessment, as described herein.

Straight algebra can be applied here, as with other yield theories fromthe field of solid mechanics, to calculate the constants, alpha (α) andbeta (β), from simple laboratory tests. The new Triaxial Leak Criterionis similar to the Drucker-Prager yield criterion, which is athree-dimensional, hydrostatic-pressure-dependent model for analyzingstresses, deformation, and failure in materials such as soils, concrete,and polymers. The Coulomb theory is a two-dimensional subset of theDrucker-Prager criterion and has been applied to model the stresses insand, soils, concrete and similar materials.

The Coulomb theory includes two material constants: angle of internalfriction (ϕ) and cohesion (c). When compared to the connection constantsof a threaded connection, the angle of internal friction (ϕ) isanalogous to alpha (α), which is associated with the geometry of thethreads. The cohesion (c) is analogous to beta (β), which is associatedwith the components of the seal, including the effects of make-upinterference and thread compound (for standard API connections) andmetal-to-metal seals or elastomer rings with lubricants or othermaterials (for non-API proprietary connections). Applying the Coulombfailure criterion in the context of the Triaxial Leak Criterion providesa formula for the constants, alpha (α) and beta (β), in terms of theCoulomb constants:

$\alpha = \frac{2\sin\phi}{\sqrt{3}\left( {3 - {\sin\phi}} \right)}$$\beta = {\frac{6c\cos\phi}{\sqrt{3}\left( {3 - {\sin\phi}} \right)}.}$

Alpha (α) is a function of the angle (ϕ) with no dependence on thecohesion (c). Beta (β) is a function of both the angle (ϕ) and thecohesion (c).

The determination of the connection constants, alpha (α) and beta (β),can be simplified by setting a number of boundary conditions which, inpractice, are useful for designing experimental tests for measuring thevalues for alpha (α) and beta (β) for a particular connection. Asdescribed herein, simple laboratory leak tests can be performed todetermine the values for the connection constants.

A FIRST EXAMPLE EXPERIMENT involves placing a threaded connection in auniaxial testing state, which is characterized by an axial loadonly—with no internal or external pressure. The axial load includestension or compression. Thus, the experiment involves two tests. First,exerting an axial force in tension on the threaded connection andmeasuring the axial tension leak stress (σt) at which leak occurs.Second, exerting an axial force in compression on the threadedconnection and measuring the axial compression leak stress (σc) at whichleak occurs. In each test, a small but insignificant pressure may beused, to measure or otherwise sense when leak occurs.

The differences in the uniaxial leak stresses generate asymmetry, aspredicted by the Drucker-Prager model. The uniaxial asymmetry ratio (m)can be expressed as:

$m = {\frac{\sigma_{t}}{\sigma_{c}}.}$

In the context of leak testing, a leak will start when the uniaxialstress (σz) reaches the axial tension leak stress (σt) or the axialcompression leak stress (σc). Because the radial stress and hoop stressare zero (since the internal and external pressures are zero), andapplying the Leak Load Equation, the formulas for the connectionconstants can be expressed as:

σ_(t)(1+α/3)=β

σ_(c)(1−α/3)=β.

Solving for the connection constants:

$\alpha = {{3\frac{\sigma_{c} - \sigma_{t}}{\sigma_{c} + \sigma_{t}}} = {3\frac{1 - m}{1 + m}}}$$\beta = {{2\frac{\sigma_{c}\sigma_{t}}{\sigma_{c} + \sigma_{t}}} = {{2\frac{\sigma_{t}}{1 + m}} = {2{\frac{m\sigma_{c}}{1 + m}.}}}}$

As shown, the constants, alpha (α) and beta (β), can be calculated basedon an experiment that measures the axial tension leak stress (σt) andthe axial compression leak stress (σc). This is fully analogous to thesimple uniaxial tension test for determining yield strength (Y) for thepipe body, except two tests are required for the threaded connectionbecause there are two constants.

The uniaxial asymmetry ratio (m) can be expressed as:

$m = {\frac{1 - \left( {\alpha/3} \right)}{1 + \left( {\alpha/3} \right)}.}$

Consideration of this formula for the uniaxial asymmetry ratio (m)demonstrates the role of alpha (α) and beta (β). If alpha is zero, thenthe ratio (m) equals one and the uniaxial leak limits for tension andcompression are equal. (When alpha equals zero in the Triaxial LeakCriterion, the effect of the mean normal stress (P) is zero, and the vonMises stress equals beta). As alpha increases for a given beta, theaxial tension leak stress (σt) decreases and the axial compression leakstress (σc) increases. For a given alpha, the two uniaxial leak stressescan only increase if beta increases.

These relationships confirm that the constant alpha (α) represents aninternal property (namely, the geometry and behavior of the threads),and the constant beta (β) represents an external property (the sealelements, including makeup).

For a threaded connection, the constant alpha (α) is a dimensionlessstress ratio that characterizes the thread behavior, as well as thetaper, to transfer the load and displacement across the threads. In thisaspect, the constant alpha (α) is a measure of the leak resistanceability of the threads and the taper. Accordingly, as used herein, thenew connection constant alpha (α) is referred to as the Thread Modulus.

The constant beta (β) is related to the ability of the seal to resistleak. According to the equations above: when beta equals zero, both theaxial tension leak stress (σt) and the axial compression leak stress(σc) equal zero. In other words, when there is no beta (i.e., no seal toresist leak), then no axial load can be supported by the connectionwithout resulting in leak. In this aspect, the constant beta (β) is ameasure of the leak resistance ability of the seal. Accordingly, as usedherein, the new connection constant beta (β) is referred to as theMakeup Leak Resistance.

A SECOND EXAMPLE EXPERIMENT for testing a threaded connection involvesplacing a threaded connection in a pressurized testing state, which ischaracterized by known internal and external pressures—with zero axialload. As with the uniaxial tests, this experiment involves two tests (todetermine the two connection constants). First, exerting an externaltest pressure (Po) on the threaded connection is used to measure theexternal leak pressure (Pb) at which external leak occurs. Second;exerting an internal test pressure (Pi) is used to measure the internalleak pressure (Pa) at which internal leak occurs.

For the external leak test, the axial load is zero, the internalpressure (Pi) is zero, and the external pressure (Po) is equal to theexternal leak pressure (Pb) at the outer radius (b). With theseconditions, the Leak Load Equation after re-arranging terms becomes:

${P_{b}\left( {1 - \frac{\alpha}{3}} \right)} = {\beta\frac{b^{2} - a^{2}}{2b^{2}}}$

where the internal radius (a) and external radius (b) refer to thedimensions of the threaded connection.

For the internal leak test, the axial load is zero, the externalpressure (Po) is zero, and the internal pressure (Pi) is equal to theinternal leak pressure (Pa) at the inner radius (a). With theseconditions, the Leak Load Equation after re-arranging terms becomes:

${P_{a}\left( {K + \frac{\alpha}{3}} \right)} = {\beta\frac{b^{2} - a^{2}}{2a^{2}}}$

where K is a geometric quantity based on the inner radius (a) and theouter radius (b), according to the equation:

${K = {\frac{1}{2}\sqrt{1 + {3\frac{b^{4}}{a^{4}}}}}}.$

Defining a pressure asymmetry ratio (n) is useful in solving for alpha(α) and beta (β) in the leak pressure relations above (for Pa and Pb).The pressure asymmetry ratio (n) is based on the internal leak pressure(Pa) at which leak occurs, and the external leak pressure (Pb) at whichleak occurs, according to the equation:

${n = {K\frac{a^{2}P_{a}}{b^{2}P_{b}}}}.$

Using the pressure asymmetry ratio (n), alpha (α) and beta (β) can bedetermined. The Thread Modulus(α) is calculated according to theequation:

${\alpha = {3K\frac{1 - n}{K + n}}}.$

The Makeup Leak Resistance (β) is calculated according to the equation:

${\beta = {{2{P_{b}\left( \frac{n\left( {K + 1} \right)}{K + n} \right)}\left( \frac{b^{2}}{b^{2} - a^{2}} \right)} = {2{P_{a}\left( \frac{K\left( {K + 1} \right)}{K + n} \right)}\left( \frac{a^{2}}{b^{2} - a^{2}} \right)}}}.$

As described herein, the Thread Modulus (α) and the Makeup LeakResistance (β) can be measured objectively by experimentation—involvingonly two tests. The constants (α, β) for a particular threadedconnection do not change. Therefore, the constants (α, β) measured usingthe pressurized testing state will be identical to the constants (α, β)measured using the uniaxial testing state. Other test conditions may beapplied to a threaded connection, resulting in the same measurements forthe constants (α, β).

A threaded connection can withstand differential pressures higher thancurrent API ratings because of the effects of hydrostatic pressure.Internal and external pressures close the connection and “energize”(enhance) the seal; axial loads open or “dilate” the connection andreduce the seal. Formulated from the new Triaxial Leak Criterion, a leakline and a leak circle are also introduced for assistance in tubulardesign, to help designers easily and quickly identify critical loads forleak and, together with the leak safety factor plot versus depth (seeFIG. 3 ) for one or more critical load cases, determines the locationwhere leak is most likely to occur along the string.

The leak line and leak circle are more comprehensive and more accuratethan existing leak envelopes because the new Triaxial Leak Criterionapplies to all combinations of internal pressure, external pressure, andaxial tension or compression—as opposed to the simple load combinations(without backup pressure) derived from the currently used lab tests orsimulations using finite-element analysis. Additionally, the new leakline and leak circle plots can be easily developed for both APIconnections and non-API proprietary connections.

LEAK LINE: The Triaxial Leak Criterion is written in the classical formof a linear equation; namely, y=mx+b, where m is the slope of the lineand b is the y-axis intercept. Accordingly, on a graph where thehorizontal or x-axis represents values for mean normal stress (P) andthe vertical or y-axis represents values for von Mises stress (σ_(VM)),the slope of the leak line 200 is alpha (α, the Thread Modulus) and they-intercept is beta (β, the Makeup Leak Resistance), as shown in FIG. 1.

FIG. 1 is a graphical representation of the Triaxial Leak Criterion 25for a particular threaded connection. The leak line 200 is plottedaccording to the Triaxial Leak Criterion 25, which is displayed in FIG.1 for reference.

The Thread Modulus (α) and the Makeup Leak Resistance (β) can bedetermined through experimentation, as described herein. These constants(α, β) do not change. The leak line 200 for a threaded connection willalways have a slope equal to the Thread Modulus (α) and a y-interceptequal to the Makeup Leak Resistance (β). In general, the leak risk ishigh for data points plotted on the graph that appear near or above theleak line 200; the leak risk is low for points significantly below theleak line 200.

In practice, the same threaded connection is used for all pipe joints ina well string. This minimizes the risk of error in the field duringassembly. Accordingly, a tubular typically consists of the same OD, ID,grade (yield strength), and threaded connection along its entire length.Sometimes a tubular has two or more sections, wherein each section hasthe same parameters (OD, ID, grade (yield strength), and threadedconnection) throughout the section.

The leak line 200 (and leak circle, described herein) apply to a sectionwhich has the same parameters. Accordingly, the Thread Modulus (α) andthe Makeup Leak Resistance (β) are the same at all points in thesection. Therefore, along the length of a section of the string, theonly variables are the set of conditions—internal pressure (Pi),external pressure (Po), and axial force (Fz)—for a given load case.

A load case, as used herein, refers to a set of conditions (pressuresand forces) that are expected to occur or may occur in the field duringthe lifetime of the well. Different load cases are based on differentload types, such as evacuation or gas kick. A load case includes aprofile of conditions—internal pressure (Pi), external pressure (Po),and axial force (Fz)—at each point (depth, z) along the string (orsection of a string). Those load values can be used to calculate themean normal stress (P) and the von Mises stress (σ_(VM)) at each depth(z) along the string, for each load case.

For example, in FIG. 1 , the locus of loads 401 on the graph is a plotof the pair of values (P, σ_(VM)) at every depth (z) along a string fora first load case. The locus of loads 402 shows the values for a secondload case; and so on. A locus of loads may be plotted for severaldifferent load cases, as shown in FIG. 1 , which provides the designerwith a visual tool for assessing the relative leak risk among differentload cases.

Each locus of loads is plotted as a continuous line because a connectioncan be situated anywhere along the tubular string. For analyticalpurposes, the fact that the threaded connections are spaced apart is notimportant. The precise location of each threaded connection is typicallynot part of the well planning process because the actual location, inthe field, can and will vary.

The curvilinear line for each locus of loads represents the relativeleak risk of a particular threaded connection at each location along thestring, relative to the leak line 200. If the locus of loads for anyload case is near or above the leak line 200, then the leak risk forthat load case is high for a string that uses that particular threadedconnection. The designer may also refer to the plot of the leak safetyfactor versus depth, for that load case, to determine the location (orlocations) along the string where the leak risk is high.

If the leak risk for any load case is higher than acceptable, the welldesigner may select a second threaded connection for use; and thenrepeat the steps of calculating and plotting the pairs of values (P,σ_(VM)) for various load cases. Note: the leak line will be differentbecause the second threaded connection has its own unique Thread Modulus(α) and Makeup Leak Resistance (β) (and the outer diameter of the secondthreaded connection may also be different). In this aspect, the leakline graph shown in FIG. 1 provides a tool for designers to evaluatedifferent threaded connections with different load cases.

LEAK CIRCLE. Like the leak line 200 shown in FIG. 1 , the leak circle600 shown in FIG. 2 applies to a string of pipe with the same threadedconnection. The graph in FIG. 2 is similar to the von Mises circle,except the formulas for each axis have been adjusted in accordance withthe new Triaxial Leak Criterion 25. If the von Mises circle axes wereused without adjustment, then a different circle radius for each depthand each load case would be required.

The fluid pressures (Pi, Po) acting on a threaded connection vary withdepth (z). The Leak Load Equation is an expression of the Triaxial LeakCriterion which includes internal and external pressures (Pi, Po). Byassuming that the pressures (on the right side of the Leak LoadEquation) at each depth (z) along the string can be approximated for allload cases by an equivalent fluid density at depth (z), and by using theinternal and external densities (pi, po), the Leak Load Equationbecomes:

${\sqrt{(\Delta)^{2} + \left( {\Delta\sigma_{z}} \right)^{2}} + {\frac{\alpha}{3}\left( {\Delta\sigma_{z}} \right)}} = {{\alpha z\frac{{b^{2}\rho_{o}} - {a^{2}\rho_{i}}}{b^{2} - a^{2}}} + \beta}$

where Δ

is the differential pressure and Δσ_(z) is the excess axial stress. Theright side of the above equation defines the leak resistance at death(z):

${L_{z} = {{\alpha z\frac{{b^{2}\rho_{o}} - {a^{2}\rho_{i}}}{b^{2} - a^{2}}} + \beta}}{\text{“Leak   Resistance   at   Depth”}\text{“z”}}$

which is displayed (16) on FIG. 2 for reference. Dividing the precedingequation by (Lz) on both sides provides the dimensionless result:

${\sqrt{\left( {\Delta/L_{z}} \right)^{2} + \left( {\Delta\sigma_{Z}/L_{z}} \right)^{2}} + {\frac{\alpha}{3}\left( {\Delta\sigma_{z}/L_{z}} \right)}} = {1.}$

Completing the square and re-arranging terms provides the formulas forthe axes for the graph in FIG. 2 ; namely, Δ

* for the y-axis and Δσ_(z)* for the x-axis:

Δ * = ( 1 - ( α / 3 ) 2 ) ⁢ Δ𝒫 L z ⁢ Δσ z * = ( 1 - ( α / 3 ) 2 ) ⁢ Δ ⁢ z Lz

where Δ

* is the normalized differential pressure and Δσ_(z)* is the normalizedexcess axial stress. These normalized values are dimensionless becauseof (Lz). These values are described as ‘normalized’ because, compared toΔ

and Δσ_(z), the normalized values include the effect of the ThreadModulus (α) and the Makeup Leak Resistance (β).

Values for normalized differential pressure (Δ

*) are displayed on the y-axis of the graph in FIG. 2 . Values fornormalized excess axial stress (Δσ_(z)*) are displayed on the x-axis.The normalized differential pressure equation 18 is displayed in FIG. 2for reference. The normalized excess axial stress equation 19 is alsodisplayed in FIG. 2 for reference. Then,

(Δ

*)²(Δσ*_(z)+α/3)²=1

is the formula for the leak circle 600, which is centered on the x-axisat Δ

*=0 and Δσ_(z)*=−α/3, and has a radius equal to one, as shown in FIG. 2.

If alpha equals zero, the leak circle reduces to the von Mises circle,which has a radius of one and is centered at the origin of the axes,which are ΔP* and Δσ_(z)* of the leak circle 600 as expressed above.

FIG. 2 is a graphical representation of the leak circle 600 at aparticular depth (z) along the string. Each depth (z) along the stringwill have its own leak circle. Each load case involves the same threadedconnection and has a pair of values (ΔP*, Δσ_(z)*), calculated using theequations above. Plotting the pair of values on the graph will display aload case point 701, which represents the relative leak risk at depth(z) for a particular load case (and a string with a particular threadedconnection) relative to the leak circle 600.

The other load case points represent the pair of values (ΔP*, Δσ_(z)*)for other load cases. The plot in FIG. 2 shows six different load casesrelative to the leak circle 600. This graph helps the designer selectthe best threaded connection to withstand all load cases withoutleaking.

The leak risk is low for load case points plotted significantly insidethe leak circle 600; points near or outside the leak circle 600 have ahigh leak risk. The closer a point is to the leak circle 600, the higherthe risk of leak (for that load case, and connection).

LEAK SAFETY PLOT: If one or more load case points are near the leakcircle 600 in FIG. 2 (and/or one or more locus of loads 401 is near theleak line 200 in FIG. 1 ), the designer may select one or more loadcases for additional scrutiny. FIG. 3 is a plot of the leak safetyfactor (SF_(Leak)) on the x-axis versus depth (z) on the y-axis for aparticular string. The leak safety equation 30 is displayed forreference.

FIG. 3 includes a line representing a threshold value 500. Typically,the threshold value 500 represents a leak safety factor that is equal toor greater than one. An additional ‘design factor’ may be selected bythe designer, as an additional margin of safety. For example, a designeror operator may specify a threshold value 500 equal to 1.25 (which isthe number typically used for the pipe body).

One or more load cases may be selected for display on the leak safetyplot shown in FIG. 3 . Each string will have its own leak safety plot.For each load case selected, the value of the leak safety factor(SFLeak) can be calculated for each depth (z) along the string andplotted, as shown in FIG. 3 . Five different load case lines are shown.

The closer the line is to the threshold value 500, the higher the leakrisk. The load case lines provide a visual representation of therelative leak risk, which may include load cases that do not meet thethreshold value 500 for leak safety. For example, the load case line 710crosses the line representing the threshold value 500, indicating a highrisk of leak. The leak safety plot also displays the depth at which leakis expected to occur for a load case. Load case line 710, for example,is expected to leak at or near a critical depth (zc).

The load case lines also provide a visual representation of the leakrisk relative to other load cases. For example, load case line 702indicates a higher safety factor compared to load case line 710 at alldepths. In this aspect, the graph may be used for a visual assessment ofa plurality of load cases, beginning with a first load case and firstload case line, and then subsequently evaluating and plotting other loadcase lines.

Just as one characteristic constant (yield strength, Y) has been usedhistorically to evaluate the mechanical limits of the pipe-bodymaterial, the two characteristic connection constants describedherein—the Thread Modulus (α) and the Makeup Leak Resistance (β)— may beused to evaluate the mechanical limits for leak of threaded connections.Without the new Triaxial Leak Criterion described herein, welldesigners, operators, and connection suppliers will continue to use: (a)expensive testing methodology and FEA to qualify threaded connections,and (b) approximate safety factors, when selecting connections forspecific applications. Neither (a) nor (b) currently considerhydrostatic dependence of connection performance. The new Triaxial LeakCriterion with two constants introduces the dependence on hydrostaticpressure, which is fundamental and different from all prior connectortheories. The new Triaxial Leak Criterion has a key assertion—mechanicalfailure in all threaded connectors depends on hydrostatic pressure.

The technology disclosed herein may be used to reduce qualificationtesting and computer stress analysis (FEA) of threaded connections—withmore efficient extrapolation across different connection types andsizes. The technology may also facilitate the optimized selection andplacement of lower-cost threaded connections, resulting in structuralefficiencies and cost savings. The technology may also facilitate thedevelopment and design of new types of threaded (and non-threaded)connections to optimize the connection constants, alpha (α) and beta(β). Further, the technology may facilitate the development and use ofnew leak safety factors for connections, which will lead to morereliable and cost-efficient well designs. This technology is applicableto not only well designs, but also to any application in any industrythat requires efficient performance of conduits with connections thathave separable male-female parts (not welded) and that seal betweeninternal and external environments. Even for non-sealing applications,such as solid (or non-solid) rods or columns with threaded connections,the new Triaxial Leak Criterion with dependence on the mean normalstress applies to structural failure of the threaded connections, suchas from pullout or thread fracture.

Connection Stresses at the Pin-Box Interface: The leak behavior of athreaded connection under triaxial stress is fully described by thetriaxial connection leak criterion, as described herein.

σ_(VM) =−αP+β  “Triaxial Leak Criterion 25”.

For the Triaxial Leak Criterion 25, the von Mises stress (σ_(VM)) is alinear function of the mean normal stress (P) in which the slope isalpha (α, the Thread Modulus) and the y-intercept is beta (β, the MakeupLeak Resistance). FIG. 1 is a graphical representation of an exampleleak line 200 plotted according to the Triaxial Leak Criterion 25 for anexample threaded connection.

For a pipe body with no bending, the von Mises stress (σ_(VM)) is equalto the yield strength Y of the pipe material. In practice, the yieldstrength Y is determined from a uniaxial tensile test on a small rodspecimen until it yields. For both types of yield (e.g., burst andcollapse) the von Mises stress (σ_(VM)) is at its maximum at the innerdiameter (ID). The leak must pass through the pipe body at the IDwhether it originates internally or externally.

For threaded connections, the pin and box are offset axially, whichmeans that an axial force F will generate a thread shear stress at thepin-box interface.

FIG. 4 is a cross-sectional view of an example tubing or casing coupledconnection 400 (e.g., API or premium) having a radial interference seal.This view of an example connection 400 includes an upper pin 410, alower pin 412, and a box 420. The conditions, as shown, include theaxial force (F), internal pressure (Pi), and external pressure (Po),each of which is known or can be measured or calculated. A_(pin)represents the cross-sectional area of the pin 410, 412 which is exposedto the internal pressure (Pi). A_(box) represents the cross-sectionalarea of the box 420 which is exposed to the external pressure (Po).

The radius (a) is correlated with the inner diameter (ID) of thethreaded connection, where the calculations were conducted for a pipebody with no bending. The radius (c) is correlated with the outerdiameter of the connection. The radius (b) represents the radiusassociated with the pin-box interface. In some example implementations,the pin-box interface radius (b) is correlated with the pitch diameter(i.e., the diameter where the thread thickness is equal to the spacebetween the threads). For some threads, the pitch diameter coincideswith the geometric center of the thread flank (the sloping side of thethread). In some implementations, the pin-box interface radius (b) isequal to the radius on the pitch line at a specified location (distancefrom pin end) when the connection is hand-tight. The pitch diameter is aknown value provided by the manufacturer or it can be reasonablyestimated. For example, for API connections, the pitch diameter isspecified at the location L1 for API 8-round or L7 for API buttress.

FIG. 5 is a cross-sectional view of an example drill string connection450 having an axial interference shoulder seal. This view of an exampleconnection 450 includes a pin 510 and a box 520. Ai represents thecross-sectional area of the pin 510 which is exposed to the internalpressure (Pi). Ao represents the cross-sectional area of the box 520which is exposed to the external pressure (Po).

There are three stress mechanisms common to all tubular threadedconnections, including the example connections 400, 450 shown in FIG. 4and FIG. 5 , respectively.

First; the seal design impacts the stresses generated in a threadedconnection. The Triaxial Leak Criterion 25 for connections,

α_(VM) =−αP+β

assumes that the seal and makeup behavior is completely characterized bythe two constants, alpha and beta, which are different for differentconnections. The actual sealing mechanism is complex and varies,including, for example, thread seals, elastomer seals, or metal-to-metalseals (for connections like the example connection 400 in FIG. 4 ) orrotary shoulder seals (for connections like the example drill stringconnection 450 in FIG. 5 ).

Second; because of the seal, hydrostatic pressure is a key driver forevaluating leak resistance. Without a seal, the pressures (internal andexternal) would penetrate the threads and load the pin and box equallyon all surfaces. When a seal is present, even an open-ended connectorsubmerged in a pressure chamber would experience shear.

Finally; because the pin and box are offset axially, the axial force Fwill generate thread shear. Thread shear generates dilatancy associatedwith flank angles, meaning the connection opens, decreasing leakresistance. The dilatancy behavior is similar to that in sand and soilswhich dilate when sheared. Even without an axial force F, the internalpressure Pi alone generates a pin-end force, as shown in FIG. 4 and FIG.5 .

For axisymmetric stresses with longitudinal thread shear stress (τ)(tau), the von Mises stress (σ_(VM)) is calculated from the expression:

$\sigma_{VM} = {\sqrt{{\frac{1}{2}\left\lbrack {\left( {\sigma_{r} - \sigma_{\theta}} \right)^{2} + \left( {\sigma_{\theta} - \sigma_{z}} \right)^{2} + \left( {\sigma_{z} - \sigma_{r}} \right)^{2}} \right\rbrack} + {3\tau^{2}}}.}$

For casing connections, like the example connection 400 in FIG. 4 , theradial stress (σr) and the hoop stress (σθ) at the pin-box interfaceradius (b) are determined using constants A and B from the Lameequations:

${\sigma_{r} = {A - \frac{B}{b^{2}}}}{\sigma_{\theta} = {A + \frac{B}{b^{2}}}}{{A = \frac{{a^{2}P_{i}} - {c^{2}P_{o}}}{c^{2} - a^{2}}},{B = \frac{a^{2}{c^{2}\left( {P_{i} - P_{o}} \right)}}{c^{2} - a^{2}}}}$

and it follows that,

$\frac{\sigma_{r} + \sigma_{\theta}}{2} = {A = \sigma_{n}}$

where A or σn is the neutral axial stress, as defined by Lubinski, andrepresents the mean normal stress in the connection cross-section.

Using the equations above for the von Mises stress (σ_(VM)) and the meannormal stress (σn), a new Leak Criterion with Thread Shear 425 in someimplementations is expressed as:

${{\sqrt{\left( {\sqrt{3}\Delta{p\left( \frac{c^{2}}{c^{2} - a^{2}} \right)}\frac{a^{2}}{b^{2}}} \right)^{2} + \left( {\Delta\sigma_{z}} \right)^{2} + {3\tau^{2}}} + {\frac{\alpha}{3}\Delta\sigma_{z}}} = {{{- \alpha}\sigma_{n}} + \beta}}.$

In practice, as described herein, a designer can approve a threadedconnection if the leak safety factor based on von Mises stress withthread shear is greater than a threshold value, which is typically anumber equal to or greater than one. An additional ‘design factor’ maybe selected by the designer (or imposed by an operator) as an additionalmargin of safety, in order to account for manufacturing tolerances andthe like. For example, a designer or operator may specify 1.25 for thethreshold value.

The Leak Criterion with Thread Shear 425 generates a circle when alphais zero and the shear stress (tau) is zero. When plotted on a graph inwhich the abscissa value (along the x-axis) is the effective stress(Δσz=σz−σn) and the ordinate value (along the y-axis) is thedifferential pressure (Δp) quantity in the bracket, the radius of thecircle is beta.

For a non-zero alpha and a shear stress (tau) of zero, the LeakCriterion with Thread Shear 425 generates a displaced circle with itscenter located at (negative-alpha divided by 3) on the dimensionlessabscissa. The circle radius depends on the neutral axial stress (σn).Accordingly, the Leak Criterion with Thread Shear 425, applied over aseries of threaded connections, would generate a series of displacedcircles each having a different radius for each depth (z).

When the neutral axial stress (σn) is replaced with the equivalentexpression in terms of differential pressure (Δp) and backup pressure,the Leak Criterion with Thread Shear 425 generates a customary skewedellipse when plotted on axes of differential pressure (Δp) and axialstress (σz). For external leak with a backup pressure (Pi) the abscissabecomes (αz+Pi), which is like pipe collapse and demonstrates the dualand equal dependence on axial stress (σz) and backup pressure (Pi).

For the square-root quantity in the Leak Criterion with Thread Shear425, the leak rating (Δp) decreases as the effective stress (Δσz=αz−αn)or the shear stress (tau) increases. This relationship means that leakis derated for axial force (F), backup pressure (Pi), and thread shearstress (tau). The API equation for internal leak does not include any ofthese effects.

The axial stress (σz) at any location in a threaded connection is notdetermined from axial force (F) alone. The internal pressure (Pi) exertsforces on the internal surfaces of the pin as shown in FIG. 4 and FIG. 5. The external pressure (Po) exerts forces on the external surfaces.

The axial force (Fb) at the mid-point of the box may be expressed as:

F _(b) =F+P _(i) A _(pin) −P _(o) A _(box)

where A_(pin) represents the exposed end area of the pin and A_(box)represents the exposed end area of the box.

The average axial stress (σz) in the thread region may be expressed as:

$\sigma_{z} = {\frac{F - {P_{o}A_{box}}}{A_{con}} = \frac{F_{b} - {P_{i}A_{pin}}}{A_{con}}}$

where A_(con) represents the cross-sectional area of the connection. Theconnection area may be calculated using A_(con)=π(c²−a²).

The axial force along the surface of the pin varies from tension (F) atthe pipe end (e.g., the lower end of the lower pin 412, as shown in FIG.4 ) to compression (— Pi A_(pin)) at the nose end of the pin (e.g., theinternal surface of the lower pin 412). This change in force along thepin equals [F−(−Pi A_(pin))], so the thread shear stress (tau) along theengaged thread length (L_(t)) may be expressed as:

$\tau = {\frac{F + {P_{i}A_{pin}}}{A_{t}} = \frac{F_{b} + {P_{o}A_{box}}}{A_{t}}}$

where A_(t) is the engaged thread area over which the shear stress (tau)acts. The engaged thread area may be calculated using A_(t)=2πbL_(t),where the engaged thread length (L_(t)) is a known value provided by themanufacturer or it can be measured or estimated. As expressed in theequation above, the shear stress (tau) acting on the pin is equal to theshear stress acting on the box. The force along the surface of the boxvaries from tension (i.e., a box force F_(b) acting at the box midpoint)to compression (−Po A_(box)) at the box end. The axial forces acting onthe pin and box generate the shear stress (tau) between the pin and box.

For known values of force (F), internal pressure (Pi), and externalpressure (Po), the three normal stresses and the thread shear stress(tau) may be calculated, starting with the Lame equations above.

The von Mises stress (σ_(VM)), as described above, may be calculatedfrom the expression:

$\sigma_{VM} = {\sqrt{{\frac{1}{2}\left\lbrack {\left( {\sigma_{r} - \sigma_{\theta}} \right)^{2} + \left( {\sigma_{\theta} - \sigma_{z}} \right)^{2} + \left( {\sigma_{z} - \sigma_{r}} \right)^{2}} \right\rbrack} + {3\tau^{2}}}.}$

The mean normal stress (P), as described above, may be calculated fromthe expression:

$P = {\frac{\sigma_{r} + \sigma_{\theta} + \sigma_{z}}{3}.}$

Applying the Triaxial Leak Criterion 25,

σ_(VM) =−αP+β

the values of the two constants, alpha and beta, are needed in order tocalculate the connection leak safety factor (SF_(C)):

${SF}_{c} = \frac{{{- \alpha}P} + \beta}{\sigma_{VM}}$

in which the numerator represents the working limit, and the denominatorrepresents the working stress. In practice, as described herein, adesigner can approve a threaded connection if the leak safety factor isgreater than a threshold value, which is typically a number equal to orgreater than one. An additional ‘design factor’ may be selected by thedesigner (or imposed by an operator) as an additional margin of safety,in order to account for manufacturing tolerances and the like. Forexample, a designer or operator may specify 1.25 for the thresholdvalue.

The two constants, alpha and beta, can be determined using one or moretests in which a number of values and conditions are known (or readilymeasured).

First Test Method: For example, two leak tests conducted on a knownconnection exposed to uniaxial force (F), no internal pressure (Pi=0),and no external pressure (Po=0) can be used to determine the constants,alpha and beta.

Using an API 7-inch 35-ppf N80 LTC connection as an example case, thefollowing data are known: inner radius (a)=3.002 inches; pin-boxinterface radius (b)=3.452 in.; outer radius (c)=3.9375 in.; engagedthread length (L_(t))=3.296 in.; exposed pin area (A_(pin))=7.164 in²;and exposed box area (A_(box))=11.023 in². Finite-element analysis (FEA)predicts that the N80 connection will open sufficiently to leak when theforce in tension (T)=620 kips (i.e., 620,000 lbf) or when thecompression (C)=690 kips. Note: the difference between the tensile andcompressive limits demonstrates the hydrostatic effect (even when thereis no applied internal or external pressure). The mean normal stress(P), using the expression above, when only axial stress (σz) is present,may be calculated using the expression, P=σz/3, which furtherdemonstrates that axial stress alone is sufficient to cause thehydrostatic effect.

The difference between the uniaxial tensile and compressive limits alsodemonstrates that the von Mises stress equation alone, with only asingle constant, would be insufficient to characterize connection leakrisk because the von Mises equation requires the uniaxial limits to beequal in magnitude. FEA analysis confirms the difference in limits. TheTriaxial Leak Criterion 25 demonstrates that two constants are requiredto characterize a connection leak risk.

A uniaxial asymmetry ratio (m) can be used to determine the constants,alpha and beta.

$m = {\frac{\sigma_{T}}{\sigma_{C}} = \frac{T}{C}}$

where the tensile stress (σ_(T)) equals the tension (T) divided by thecross-sectional area (A_(con)) of the LTC connection, and thecompressive stress (σ_(C)) equals the compression (C) divided by thecross-sectional area (A_(con)) of the LTC connection.

In laboratory testing, a uniaxial leak test can be conducted with eithera small amount of pressure or with some non-mechanical sensor. Foruniaxial test conditions with a small pressure for leak detection,internal pressure (Pi) is considered zero and external pressure (Po) isconsidered zero; therefore, the differential pressure (Δp) is zero andthe neutral axial stress (σn) is zero.

For uniaxial tension, the tensile leak limit is reached when theuniaxial stress (σz) reaches the tensile stress (σ_(T)). For tension,the Leak Criterion with Thread Shear 425 reduces to:

${\sigma_{T}\left\lbrack {\sqrt{1 + {3\left( \frac{A_{con}}{A_{t}} \right)^{2}}} + \frac{\alpha}{3}} \right\rbrack} = {\beta.}$

For uniaxial compression, the compressive leak limit is reached when theuniaxial stress (σz) reaches the compressive stress (negative (−σ_(C))).For compression, the Leak Criterion with Thread Shear 425 reduces to:

${\sigma_{C}\left\lbrack {\sqrt{1 + {3\left( \frac{A_{con}}{A_{t}} \right)^{2}}} - \frac{\alpha}{3}} \right\rbrack} = {\beta.}$

Defining a uniaxial test constant (S),

$S = \sqrt{1 + {3\left( \frac{A_{con}}{A_{t}} \right)^{2}}}$

and solving for alpha and beta, the Thread Modulus alpha (α) equals:

$\alpha = {3\frac{1 - m}{1 + m}S}$

and the Makeup Leak Resistance beta (β) equals:

$\beta = {{{\sigma_{T}\left( \frac{2}{1 + m} \right)}S} = {\left( \frac{T}{A_{con}} \right)\left( \frac{2}{1 + m} \right)S}}$or$\beta = {{{\sigma_{C}\left( \frac{2m}{1 + m} \right)}S} = {\left( \frac{C}{A_{con}} \right)\left( \frac{2m}{1 + m} \right){S.}}}$

Solving for the uniaxial asymmetry ratio (m) from the equation above foralpha,

$m = \frac{S - \left( {\alpha/3} \right)}{S + \left( {\alpha/3} \right)}$

which means the ratio (m) is between zero and one (0≤m≤1) and the ThreadModulus alpha (α) is between zero and {three times S} (0≤α≤3S) in caseswhere alpha is positive. The value of alpha may be negative for certainload conditions and certain types of connections.

For the example LTC connection, and the FEA limits for tension (T=620kips) and compression (C=690 kips), the ratio (m) is 0.899 and theuniaxial test constant (S) is 1.115. According to the equations above,the Thread Modulus alpha (α) is 0.179 and the Makeup Leak Resistancebeta (β) is 35,720 psi.

Although the values for alpha and beta were calculated based on theexample uniaxial test conditions described above, the Thread Modulusalpha (α) and the Makeup Leak Resistance beta (β) are considered to beconstant values associated with the example LTC connection, under anyconditions, including field conditions. Accordingly, the constant valuesfor alpha and beta are expected to be the same under different testconditions.

Second Test Method: In another example test, the two constants, alphaand beta, can be determined when a number of values and conditions areknown or readily measured. In this second test method, two pressuretests are used. An internal leak test is conducted on a known connectionexposed to a zero uniaxial force (F=0), no external pressure (Po=0), anda known internal leak pressure (Pi=Pa). The internal pressure (Pi)equals the leak pressure (Pa) at the inner radius because internal leakwould occur at the inner radius (a). Under these conditions, the axialstress (σz) in the thread region is zero.

An external leak test is conducted on a known connection exposed to azero uniaxial force (F=0), no internal pressure (Pi=0), and a knownexternal leak pressure (Po=Pc). The external pressure (Po) equals theexternal leak pressure (Pc) at the outer radius because external leakwould occur at the outer radius (c). Under these conditions, the shearstress (tau) is zero because the force is zero (F=0) and the internalpressure is zero (Pi=0).

The equations for internal and external leak can be simplified by usingand calculating a number of pressure-test constants (K₀, K₁, K₂, and K₃)which are unitless and based on the geometry of the connection:

$K_{0} = {1 - {\left( \frac{c^{2} - a^{2}}{c^{2}} \right)\left( \frac{A_{box}}{A_{con}} \right)}}$$K_{1} = {\frac{1}{2}\sqrt{{3\left( \frac{a}{b} \right)^{4}} + \left( K_{0}^{2} \right)^{4}}}$$K_{2} = {\frac{1}{2}\left( {3 - K_{0}} \right)}$$K_{3} = {\frac{1}{2}{\sqrt{1 + {3\left( \frac{c}{b} \right)^{4}} + {3\left( \frac{A_{pin}}{A_{t}} \right)^{2}\left( \frac{c^{2} - a^{2}}{a^{2}} \right)^{2}}}.}}$

Recall, the Leak Criterion with Thread Shear 425 described above:

${\sqrt{\left( {\sqrt{3}\Delta{p\left( \frac{c^{2}}{c^{2} - a^{2}} \right)}\left( \frac{a^{2}}{b^{2}} \right)} \right)^{2}} + \left( {\Delta\sigma}_{z} \right)^{2} + {3\tau^{2}} + {\frac{\alpha}{3}\Delta\sigma_{z}}} = {{{- \alpha}\sigma_{n}} + {\beta.}}$

Using the pressure-test constants (K₀, K₁, K₂, and K₃), the LeakCriterion with Thread Shear 425 for internal leak at the inner radius(a) reduces to:

${P_{a}{\frac{2a^{2}}{c^{2} - a^{2}}\left\lbrack {K_{3} + \frac{\alpha}{3}} \right\rbrack}} = \beta$

and the Leak Criterion with Thread Shear 425 for external leak at theouter radius (c) reduces to:

${P_{c}{\frac{2c^{2}}{c^{2} - a^{2}}\left\lbrack {K_{1} - {K_{2}\frac{\alpha}{3}}} \right\rbrack}} = {\beta.}$

Whereas a uniaxial asymmetry ratio (m) was applied above for theuniaxial force test, a unilateral asymmetry ratio (n) may be appliedhere for the pressure-test conditions, where n=(a² Pa)/(c² Pc).

Using the unilateral asymmetry ratio (n), the above equations can besolved for alpha and beta:

$\alpha = {3\frac{K_{1} - {K_{3}n}}{K_{2} + n}}$$\beta = {{2P_{a}\frac{a^{2}}{c^{2} - a^{2}}\left( \frac{K_{1} + {K_{2}K_{3}}}{K_{2} + n} \right){or}{}\beta} = {2P_{c}\frac{c^{2}n}{c^{2} - a^{2}}{\left( \frac{K_{1} + {K_{2}K_{3}}}{K_{2} + n} \right).}}}$

For the example LTC connection, the equation for Leak Criterion withThread Shear 425 and the equations for alpha and beta (above) predictthat internal leak will occur when the pressure (Pa) reaches 9,975 psiand that external leak will occur when the external leak pressure (Pc)reaches 10,770 psi. Using these predicted internal leak pressure (Pa)and the external leak pressure (Pc), along with the geometry data forthe example LTC connection in the above equations, confirms that theThread Modulus alpha (α) equals 0.179 and the Makeup Leak Resistancebeta (β) equals 35,720 psi.

Finite-element analysis (FEA) predicts that the example LTC connectionwill develop internal leak when the pressure (Pa) reaches 9,942 psi,which is close to the predicted value of 9,975 psi. The API internalleak value for the example LTC connection is 11,790 psi.

FIG. 6 is a graphical representation of the predicted leak conditionsfor an example LTC connection. When plotted on a graph in which theabscissa value (along the x-axis) is the axial force (F) and theordinate value (along the y-axis) is the differential pressure(Δp=Pi−Po), the predicted leak values form an ellipse.

As shown on the graph in FIG. 6 , the solid X represents the APIinternal leak value of 11,790 psi for the example N80 connection whenthe axial force is zero (F=0).

The values for internal leak are plotted above the x-axis, where thedifferential pressure (Δp) is positive. The values for external leak areplotted below the x-axis, where the differential pressure (Δp) isnegative. The transitions between the upper and lower halves of theellipses are not smooth because the mean normal stress (σ_(n)) isdifferent for internal leak conditions versus external leak conditions.The triangles in FIG. 6 represent the values predicted by finite-elementanalysis (FEA).

The three ellipses plotted in FIG. 6 are based on the Leak Criterionwith Thread Shear 425, using the alpha and beta values that werecalculated using the formulas shown above.

The first ellipse is plotted using a solid line for internal leak (abovethe x-axis) and a dotted line for external leak (below the x-axis). Asshown in the legend, the values for internal leak were calculated usingan external pressure of zero (Po=0). The values for external leak werecalculated using an internal pressure of zero (Pi=0). The solidtriangles represent the values predicted by finite-element analysis(FEA) when the external pressure is zero (Po=0). As described above, FEApredicted that the LTC connection will open sufficiently to leak whenthe force in tension (T)=620 kips (e.g., a positive force (F) along thex-axis) or when the compression (C)=690 kips (e.g., a negative force).As shown, the first ellipse (solid line) crosses the positive y-axisvery near the solid triangle. The first ellipse (dotted line) crossesthe negative y-axis at minus 10,770 psi.

The second ellipse is plotted using a dashed line with a single dot forinternal leak (above the x-axis) and a dashed line for external leak(below the x-axis). As shown in the legend, the values for internal leakwere calculated using an external pressure of 5,000 psi. The values forexternal leak were calculated using an internal pressure of 5,000 psi.The open triangles represent the values predicted by finite-elementanalysis (FEA) when the external pressure is 5,000 psi. As shown, theopen triangle for the compression limit (along the negative x-axis) isclose to the value predicted (where the second ellipse crosses thex-axis). The open triangle for the tension limit (along the positivex-axis) is somewhat greater than the value predicted. The open trianglealong the positive y-axis is close to the value predicted. The secondellipse (dashed line) crosses the negative y-axis at minus 10,630 psi.

The third ellipse is plotted using a dashed line with two dots forinternal leak (above the x-axis) and a double-dotted line for externalleak (below the x-axis). As shown in the legend, the values for internalleak were calculated using an external pressure of 10,000 psi. Thevalues for external leak were calculated using an internal pressure of10,000 psi. The shaded triangle represents the value predicted byfinite-element analysis (FEA) when the external pressure is 10,000 psi.As shown, the shaded triangle along the positive y-axis is close to thevalue predicted. The third ellipse (double-dotted line) crosses thenegative y-axis at minus 10,450 psi.

The three ellipses plotted in FIG. 6 are off-center, as expected,because of hydrostatic pressure. As backup pressure increases, theellipse shifts to the left and the difference between the tension andcompression limits increases (i.e., the curves are further apart)because of the (αP) term in the Triaxial Leak Criterion 25:

σ_(VM) =−αP+β.

FIG. 7 is a graphical representation of the predicted failure conditionsfor the pipe-body alone (steel yield strength (Y) for the outer ellipse,shown using dashed lines) and for the connection alone (leak for theinner ellipse, shown using solid and dotted lines) for the example LTCconnection. The abscissa values (along the x-axis) represent the axialforce (F) and the ordinate values (along the y-axis) represent thedifferential pressure (Δp=Pi−Po).

For the pipe-body (the outer ellipse), the Von Mises equation is usedwith the yield strength (Y) of the N80 pipe material (Y=80,000 psi). ForΔp=0, the pipe ellipse shows that both the compressive and tensilelimits are the same, namely 814 kips (based on Y times thecross-sectional area of the example pipe body). Loads inside the pipeellipse indicate that the pipe is safe.

For the connection (the inner ellipse), the Leak Criterion with ThreadShear 425 is used to calculate the values on the ellipse. The innerellipse for the connection is mostly inside the outer pipe-body ellipse,which indicates the connection will leak before the pipe body fails.

For evaluating leak, predicted values inside the inner ellipse indicatea safe connection, whereas values outside the leak ellipse indicate theconnection will fail.

The inner ellipse for the connection is mostly inside the outerpipe-body ellipse, except for a small section in quadrant three. The twoellipses cross near the API collapse limit (shown by the X), suggestingthat the pipe-body will collapse before external leak occurs.

Although the various embodiments are described with reference topetroleum engineering and down-the-hole drilling and production for oiland gas, the methods and systems described herein may be applied in avariety of contexts. For example, the technology and solutions describedherein may be readily applied to any use or application that includes acolumn (vertical, horizontal, or otherwise) of elements having one ormore threaded connections, especially when such a column is positionedwithin any substance that may be characterized as a fluid or causingfluid-like pressures on the column, such as from cement, soil, and rockor generally from any solid that squeezes or confines the column.

FIG. 8 is a partial cross-sectional view of an example generic, premiumthreaded connection 650 which has been slightly modified to leak intension before the pipe body fails. The connection 650 selected foranalysis herein is a generic premium connection; a 9⅝-inch 40-ppf N80coupled casing connection with an internal metal-to-metal seal,buttress-type threads, and a torque shoulder. The connection 650 wasmodified for use with finite element analysis (FEA) to determine theleak constants (e.g., the thread modulus (α) and the makeup leakresistance (β)) as described herein. For example, the connection 650 wasmodified to have a reduced seal contact at make-up so the connection 650would leak under moderate to high loads before pipe yield (i.e., thepipe body von Mises stress is within the yield limit). Otherwise, thepipe body would yield before the connection leaks.

For conducting a finite element analysis (FEA) of the example premiumconnection 650, the N80 casing material was modeled using a bilinearstress-strain relationship with an ultimate tensile strength of 100 ksiat 10% strain. An elastic modulus of 30,000 ksi and a Poisson's ratio of0.3 were used. Contact interaction was modeled between the pin and thebox. Friction was not considered at the contact interface. An axialsymmetry boundary condition was applied at the coupling center, wherethe boundary was fixed along the longitudinal direction but was allowedto move radially. Pressure access was assumed; internal pressure wasapplied on the pin-shoulder interface to the seal, and external pressurewas applied in the thread region to the seal.

A finite element analysis (FEA) of the example connection 650 evaluatedeleven (11) load cases that would result in leaks. The leaking testcases 700A are tabulated in FIG. 9 including the combination of loads(e.g., pressures and forces) which, based on the FEA, would be expectedto result in leak of the example connection 650. The FEA predicted thetensile forces (F) which, given the pressures, would produce leaks atthe connection 650. For example, for leaking test case 1 in FIG. 9 , theFEA predicted that a tensile force (F) equal to 621 kips (tension ispositive and compression is negative) would produce a leak. The testcases 700A can be actual, physical tests conducted in a laboratory whichleak or FEA tests that predict leak. In either case, the tests aredesigned such that the connection leaks before the pipe-end of the testspecimen fails.

FIG. 9 also includes a number of calculated values associated with eachleaking test case 700A. For example, each differential pressure value(Δp) (=Pi−Po) 910 is calculated by subtracting the external pressure(Po) from the internal pressure (Pi). The normal stresses (radial, hoop,axial), the shear stress (tau), the Lubinski neutral stress (σ_(n)), andeach von Mises stress value (σ_(VM)) 930 were calculated using theequations discussed herein. The mean normal stress (P) is calculated asthe average of the three normal stresses, as described herein.

FIG. 9 also includes an effective pressure value (Pe) 920 associatedwith each leaking test case 700A. The value for each effective pressure(Pe) (=P−σn) is calculated by subtracting the neutral stress (σn) fromthe mean normal stress (P). For example, leaking test case 5 has a meannormal stress (P) equal to −2.523 and a neutral stress (σn) of −10. Theeffective pressure (Pe) for leaking test case 5 equals P (−2.523) minusan (−10), which equals +7.477.

FIG. 10 is a graph 900 of the leaking test cases 700A tabulated in FIG.9 . As shown, the graph 900 in FIG. 10 is a Cartesian coordinate systemin which each abscissa value along the x-axis is an effective pressure920 (Pe) (=P−σn) and each ordinate value along the y-axis is a von Misesstress 930. As shown, the plotted data values associated with eachparticular differential pressure (e.g., Δp=2, Δp=1, Δp=0, Δp=1, Δp=2)are grouped together because they are nearly overlapping. For example,starting at the left side of the plotted values, a differential pressureof Δp=2 is associated with one case; specifically, leaking test case 8in FIG. 9 , in which the effective pressure Pe is 4.897 and the vonMises stress is 27.742.

The differential pressure of Δp=1 is associated with two cases;specifically, leaking test cases 6 and 7 in FIG. 9 . Note, the plottedvalues associated with the two cases of Δp=1 are nearly overlapping.

The differential pressure of Δp=0 is associated with five cases;specifically, leaking test cases 1-5 in FIG. 9 . Again, the plottedvalues associated with these five leaking test cases are nearlyoverlapping.

As shown in FIG. 10 , the grouping of plotted values associated with theeleven leaking test cases 700A has a quadratic dependence on effectivepressure (Pe) (=P−σn) with three constants. Curve fitting refers to theprocess of constructing a curve or a mathematical function that bestfits a series of data points. In most types of curve fitting, the dataare fitted using a number of successive approximations. Linear orstraight-line fitting is a common form in which the mathematical modelis a linear expression (e.g., y=mx+b). When the data points indicate aquadratic dependence, such as those shown in FIG. 10 , the mathematicalmodel is a quadratic expression (e.g., y=ax²+bx+c) that most closelyfits the data.

The leak behavior of a threaded connection as a function of effectivepressure (Pe) (=P−αn) is described by a new quadratic leak criterion325:

σ_(VM) =δP _(e) ² −αP _(e)+β

where the first constant is called the leak path factor (δ) and theother two constants are the thread modulus (α) and the makeup leakresistance (β), as described herein. The first constant is called theleak path factor (δ) because it appears to depend on the leak flow pathand the differential pressure (Δp) or, more specifically, on internalleak versus external leak.

The three leak constants will be different for different connections.Because thread modulus (α) and leak path factor (δ) are multipliers ofthe load terms, these two constants in some cases may depend onconnection type instead of size; so those constants may be the same fora family of connections having the same type. The makeup leak resistance(β) will be different for different sizes, but it may correlate with oneor more geometric parameters for the same family of connections.

For the example leaking test cases 700A tabulated in FIG. 9 and shown onthe graph 900 in FIG. 10 , the fitting quadratic expression 330a has aleak path factor (δ) equal to 1.61 psi⁻¹, a unitless thread modulus (α)of 21.43 and a makeup leak resistance (β) of 94.37 psi. The fittingquadratic expression 330a, as shown, produces a parabola 331 thatapproximately fits the plotted data points.

Any of a variety of curve fitting models and tools may be used toperform the process of fitting a quadratic expression to the datapoints. For example, applications with graphical tools, such asMicrosoft Excel, may include one or more fitting models (e.g.,exponential, linear, logarithmic, polynomial, and the like) foranalyzing sets of data points. In some implementations, the method offitting includes plotting the data points on a graph, as shown in FIG.10 .

The quadratic leak criterion 325 can also be expressed, of course, as aleak safety factor 330, according to the equation:

${SF}_{Leak} = \frac{{\delta P_{e}^{2}} - {\alpha P_{e}} + \beta}{\sigma_{VM}}$

in which the numerator represents the working stress limit and thedenominator represents the working stress. In practice, as describedherein, a designer can approve a threaded connection (for theanticipated loads) if the leak safety factor is greater than a thresholdvalue, which is typically a number equal to or greater than one. Anadditional ‘design factor’ may be selected by the designer (or imposedby an operator) as an additional margin of safety, in order to accountfor manufacturing tolerances and the like. For example, a designer oroperator may specify a value of 1.25 or greater for the threshold value.

FIG. 11 is a graphical representation of a quadratic leak safety factor330 for an example string of threaded connections, including exampleload case lines for a number of example load cases 700 b. The quadraticleak safety factor for each string is plotted on the x-axis, versusdepth (z) on the y-axis, for each example string. Each string will haveits own leak safety plot. The graph includes a line representing thethreshold value 500 b. For each of the example load cases 700 b, thevalue of the leak safety factor can be calculated for each depth (z)along the string and plotted.

Five different load case lines are shown. The closer the line is to thethreshold value 500 b, the higher the leak risk. The load case linesprovide a visual representation of the relative leak risk, which mayinclude load cases that do not meet the threshold value 500 b for leaksafety. For example, the load case line 710 b crosses the linerepresenting the threshold value 500 b, indicating a high risk of leak.The leak safety plot also displays the depth at which leak is expectedto occur for a load case. Load case line 710 b, for example, is expectedto leak at or near a critical depth (zc).

The load case lines also provide a visual representation of the leakrisk relative to other load cases. For example, load case line 702 bindicates a higher safety factor compared to load case line 710 b (atall depths). In this aspect, this kind of graph may be used for a visualassessment of a plurality of load cases 700 b.

As used herein, the term “optimize” refers to and includes selecting athreaded connection that will withstand the assumed loads in the wellplanning or design phase. In addition, the term “optimize” also refersto and includes the management or adjustment of the loads (e.g.,pressures and forces), during the operation phase, to ensure safety andto maintain connection integrity. For example, a threaded connection mayhave been selected, delivered, or installed. The methods and systemsdescribed herein may be applied to optimize the operation of theconnection by changing or adjusting the loads that will be acting on aconnection.

Except as stated immediately above, nothing that has been stated orillustrated is intended or should be interpreted to cause a dedicationof any component, step, feature, object, benefit, advantage, orequivalent to the public, regardless of whether it is or is not recitedin the claims.

It will be understood that the terms and expressions used herein havethe ordinary meaning as is accorded to such terms and expressions withrespect to their corresponding respective areas of inquiry and studyexcept where specific meanings have otherwise been set forth herein.Relational terms such as first and second and the like may be usedsolely to distinguish one entity or action from another withoutnecessarily requiring or implying any actual such relationship or orderbetween such entities or actions.

Unless otherwise stated, any and all measurements, values, ratings,positions, magnitudes, sizes, and other specifications that are setforth in this specification, including in the claims that follow, areapproximate, not exact. Such amounts are intended to have a reasonablerange that is consistent with the functions to which they relate andwith what is customary in the art to which they pertain. For example,unless expressly stated otherwise, a parameter value or the like mayvary by as much as plus or minus ten percent from the stated amount orrange.

In addition, in the foregoing Detailed Description, it can be seen thatvarious features are grouped together in various examples for thepurpose of streamlining the disclosure. This method of disclosure is notto be interpreted as reflecting an intention that the claimed examplesrequire more features than are expressly recited in each claim. Rather,as the following claims reflect, the subject matter to be protected liesin less than all features of any single disclosed example. Thus, thefollowing claims are hereby incorporated into the Detailed Description,with each claim standing on its own as a separately claimed subjectmatter.

While the foregoing has described what are considered to be the bestmode and other examples, it is understood that various modifications maybe made therein and that the subject matter disclosed herein may beimplemented in various forms and examples, and that they may be appliedin numerous applications, only some of which have been described herein.It is intended by the following claims to claim any and allmodifications and variations that fall within the true scope of thepresent concepts.

What is claimed is:
 1. A method of evaluating leak safety, comprising:calculating a leak safety factor (SF_(Leak)) associated with a threadedconnection as a function of an effective pressure (Pe) and a von Misesstress (σ_(VM)), according to the equation:${SF}_{Leak} = \frac{{\delta P_{e}^{2}} - {\alpha P_{e}} + \beta}{\sigma_{VM}}$wherein a leak path factor (δ), a thread modulus (α), and a makeup leakresistance (β) are constants associated with the threaded connection;and approving the threaded connection if the leak safety factor isgreater than a threshold value.
 2. The method of claim 1, wherein theeffective pressure (Pe) is defined by a mean normal stress (P) minus aneutral axial stress (σn).
 3. The method of claim 1, wherein thethreshold value is a number equal to or greater than one.
 4. The methodof claim 1, further comprising: assessing a leak risk associated withthe threaded connection according to a quadratic leak criterion:σ_(VM) =δP _(e) ² −αP _(e)+β
 5. The method of claim 1, whereincalculating the leak safety factor further comprises: generating atleast three leaking test cases, each having a different differentialpressure value Δp (=Pi−Po), and each characterized by a combination ofloads expected to result in leak at the threaded connection; for each ofthe leaking test cases, calculating a subset of values comprising aneffective pressure value (Pe) and a von Mises stress value (σ_(VM)); andfitting a quadratic expression to the subset of values, wherein thefitted quadratic expression comprises values for the leak path factor(δ), the thread modulus (α), and the makeup leak resistance (β).
 6. Themethod of claim 5, further comprising: plotting on a graph the leakingtest cases, each represented by its respective effective pressure value(Pe) and its respective von Mises stress value (σ_(VM)); and fitting thequadratic expression to the plotted leaking test cases, wherein thequadratic expression when plotted on the graph produces a parabola.